(A) Let $f(x) = x^2 + 2(a - 3)x + 9$.For a quadratic $f(x) = x^2 + bx + c$ to have roots $\alpha, \beta$ such that $\alpha < k < \beta$,the condition

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Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is a correct explanation for Statement-$I$.

(A) Let $f(x) = x^2 + 2(a - 3)x + 9$.For a quadratic $f(x) = x^2 + bx + c$ to have roots $\alpha, \beta$ such that $\alpha Here,$k = 6$,so we require $f(6) $f(6) = (6)^2 + 2(a - 3)(6) + 9 $36 + 12(a - 3) + 9 $45 + 12a - 36 $12a + 9 $12a $a Thus,Statement-$II$ is the condition for Statement-$I$ to hold,and both are true.

Class 11 Mathematics 4-2.Quadratic Equations and Inequations Quadratic expressions and Position of roots

Medium

Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.
Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.

A
Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is a correct explanation for Statement-$I$.
B
Statement-$I$ is true,Statement-$II$ is true,Statement-$II$ is not a correct explanation for Statement-$I$.
C
Statement-$I$ is true,Statement-$II$ is false.
D
Statement-$I$ is false,Statement-$II$ is true.

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Class 11

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4-2.Quadratic Equations and Inequations

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Quadratic expressions and Position of roots

If the roots of ${x^2} + x + a = 0$ exceed $a$,then

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Number of integral values of $a$ for which the smaller root of the quadratic equation $x^2 - 2ax + a^2 - 4 = 0$ is smaller than $1$ and the bigger root is greater than $6$ is:

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If the roots of the quadratic equation $x^2 - 2kx + k^2 + k - 5 = 0$ are less than $5$,then in which interval does $k$ lie?

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Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

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If $b > a$,then in which interval do the roots of the equation $(x - a)(x - b) - 1 = 0$ lie?

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Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.

Similar Questions

If the roots of ${x^2} + x + a = 0$ exceed $a$,then

Number of integral values of $a$ for which the smaller root of the quadratic equation $x^2 - 2ax + a^2 - 4 = 0$ is smaller than $1$ and the bigger root is greater than $6$ is:

If the roots of the quadratic equation $x^2 - 2kx + k^2 + k - 5 = 0$ are less than $5$,then in which interval does $k$ lie?

Let $y = f(x) = ax^2 + 2bx + c$,where $a, b, c \in R$ and $a \neq 0$. If $f(x) = 0$ has imaginary roots and $4a + 4b + c < 0$,then which of the following is true?

If $b > a$,then in which interval do the roots of the equation $(x - a)(x - b) - 1 = 0$ lie?

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Statement-$I$: If the roots $\alpha, \beta$ of the equation $x^2 + 2(a - 3)x + 9 = 0$,$a \in R$ satisfy $\alpha < 6 < \beta$,then $a < -3/4$.
Statement-$II$: If $f(x) = x^2 + 2(a - 3)x + 9$,then $f(6) < 0 \implies a < -3/4$.